Tutte's invariant approach for Brownian motion reflected in the quadrant

Abstract

We consider a Brownian motion with drift in the quarter plane with orthogonal reflection on the axes. The Laplace transform of its stationary distribution satisfies a functional equation, which is reminiscent from equations arising in the enumeration of (discrete) quadrant walks. We develop a Tutte's invariant approach to this continuous setting, and we obtain an explicit formula for the Laplace transform in terms of generalized Chebyshev polynomials.

14 pages, 3 figures

References in corpus (2)

• Reflected planar Brownian motions, intertwining relations and crossing probabilities
• Counting quadrant walks via Tutte's invariant method (extended abstract)

Cited by in corpus (7)

• Integral expression for the stationary distribution of reflected brownian motion in a wedge
• Escape and absorption probabilities for obliquely reflected Brownian motion in a quadrant
• Green's Functions with Oblique Neumann Boundary Conditions in the Quadrant
• Probability of total domination for transient reflecting processes in a quadrant
• On the stationary distribution of reflected Brownian motion in a wedge: differential properties
• Asymptotics for the Green's functions of a transient reflected Brownian motion in a wedge
• Reflected Brownian Motion in a wedge: sum-of-exponential absorption probability at the vertex and differential properties